Download Real business cycles

Real business cycles
Prof. A. Pommeret
I. Jaccard
HEC Lausanne, 2003/2004
Problem set 5: Sketch of Answers
(June, 14th 2004)
You can work in groups of up to 4 persons. Your answers are due by June 1st. For
undergraduate students question 3/ of part II is optional.
Part I : Open questions (Exam 2002)
a/ How the existence of a mismatch between labor demanding agents and labor supplying
agents can be taken into account in an RBC model ?
Answer:
The standard literature being unable to explain the high and persistent level of unemployment,
some authors including Langot (1994) have introduced non-walrasian labor market feature
where the allocation of labor is led by a search process. It is assumed that there exists a wellbehaved matching function linking the number of aggregate hirings H t with the aggregate
search behavior of the agents:
H t  H 0 u t1 vt
vt gives the number of vacant jobs. This indicator represents the recruitment effort of
firms. Only vacant jobs are part of the exchange process.
- u t gives the aggregate unemployment rate in the economy. By definition it is equal to
(1  nt ) where nt is the employment level in the economy.
-
b/ What is the probability for a firm j to have a vacant job becoming filled, as a function of
the labor market tightness? What is the probability for an unemployed worker i to find a job,
as a function of the labor market tightness?
Answer:
The probability of that a vacant job becomes filled for the firm j is given by:
u 
H
qi , j  t   t ,1    t ,1  q t 
vt
 vt 
 
ut
gives the friction rate on the labor market
vt
For an unemployed worker, the transition probability from unemployment to employment is
given by:
where  t 
p i ,t 
 v 
Ht
 1, t    1, t1  p t 
ut
 ut 


c/ Give the expression of the Beveridge curve
Answer:
The labor process at the aggregate level is given by:
nt 1  (1  s)nt  (ut , vt )
where H t  H 0 u t1 vt .
The Beveridge curve concerns what is happening at the steady state when nt 1  nt . So:
s(1  u)  (u, v)  H 0 u1 v
which gives the Beveridge curve.
d/ Describe how does such a mismatch affect the household and the firm program
Answer:
The households i supplies in an inelastic way one unit of labor. The instantaneous utility U it
of the agent i at period t is:
U it   itU (Cite )  (1   it )U (Citu )
It gives in fact the utility that the agent i can hope having at period t knowing all her past.
World being Markovian, it is enough to know the state at the preceding period.  it gives the
probability of the agent to be employed at period t:
1  s if the agent is employed in t  1

 it  
 p( t 1 ) if the agent is unemployed in t  1

U (.) gives the utility the agent gets from its consumption level at time t and it is defined by:
log( Cite  t ) if the agent is employed in t

U (Citn )  
u
log( Cit ) if the agent is unemployed in t

Solving the program of agent i provides her consumption/saving schedule and its insurance
behavior. This behavior satisfies the following Bellman equation:
Vit ( Bit , it )  max Lit
with Lit given by:





Lt   i ,t log( Cite  t )  Et (1  s )Vite1 Bite 1 , I it 1  sVitu1 Bitu1 , I it 1

 eit Bit  Wijt   it  Cite   t Ait    (t  1 t )Bite 1 ( z t 1 )dz t 1


Z





 (1   i ,t ) log( C itu  t )  Et p( t )Vite1 Bite 1 , I it 1  (1  p( t ))Vitu1 Bitu1 , I it 1

 uit Bit  Ait   it  Citu   t Ait    (t  1 t )Bitu1 ( z t 1 )dz t 1
Z
it  zt ,Wi , j ,t ,  (t  1t )


with
Optimality conditions of this program are:
1 /(Cite  t )  eit
(20)
1 /(Citu  t )  uit
(21)
 it t eit  (1   it )(1   t )uit
(22)
 it (t  1 t )eit   it (1  s)  (1   it ) p( t )VBe it 1 f ( z t 1 z t )
(23)
(1   it ) (t  1 t )uit   it s  (1   it )(1  p( t )) VBuu it 1 f ( z t 1 z t )
(24)
e
Compared to the standard case, the main difference is, firstly, that the probability of being
employed in t-1 is given by the transition probability from unemployment to employment
defined earlier (see b/) and thus that labor is supplied in an inelastic way. Moreover, since the
wage is not the price which clears a walrasian labor market, the resolution of this problem
does not characterize a Pareto Optimum.
The firm's j problem
The producer j defines his production schedule by maximizing the discounted sum of his
expected profits. The recursive problem has a solution that satisfies the following Bellman
equation:
V f ( K jt , n jt , t )  max Y jt  Wijt n jt  I jt   t v jt
I jt ,v jt
   (t  1 t )V f ( K jt 1 , n jt 1 , t 1 )dz t 1
Z

 K jt 1  I jt  (1   ) K jt
subject to: 
n jt 1  (1  s )n jt  q ( t )v jt
Note that in contrast to the standard case, employment is now predetermined and thus if firms
create new jobs in t, this will only affect employment at time t+1. In addition, the unitary cost
due to a new vacant job,  t , is now entering the firm maximization problem. There exists no
increasing returns in labor in the sense that the development level of the economy does not
allow to reduce hiring costs.
While the investment behavior of the firm is not affected by the mismatch, the labor demand
can now be summarized by the following condition:
t
n jt
q ( t )
Firms providing new vacant jobs until the expected marginal value of labor being equal to the
holding of vacant jobs (free-entry condition), the above equation expresses the fact that the
firm hires until the marginal value of labor (here the profit associated with one more job) is
equal to the cost generated by having labor that is the wage payment net of the cost of holding
a vacant job.
Vnjtf  (1   )
Y jt
 Wijt  (1  s )
We assume that the wage is set through a wage bargaining process to close the model.
e/ What do you expect as far as the productivity cycle replication is concerned (and why)?
Answer:
In contrast to the standard RBC model where both employment and labor productivity
respond instantaneously to technological shocks and exhibit a discrete jump at the time of the
shock, in this setting the existence of a mismatch between firms and households will imply
that employment will be a predetermined variable. As a result, similarly to the response of
capital stock, the response of employment to technology shocks will be gradual and humped
shaped. In contrast, labor productivity, (Y/N), will instantaneously jump in response to a
technological shock and then monotically decrease to eventually converge back to the steady
state. As a result, the fact that employment is now predetermined should help us to reduce the
correlation between these two variables. Hence we should obtain a propagation mechanism
which is able to reproduce the labor productivity cycle, i.e. that the labor productivity is an
advanced indicator of employment dynamics since in such a model, a rise in labour
productivity will be followed by a progressive rise in employment.
f/ In such a model, how do you explain that unemployment and vacant jobs exhibit a too small
volatility compared to the stylized facts while labor productivity and wages are too volatile ?
Why are theoretical wage and labor productivity moments so close?
Answer:
The strong unemployment persistence that is generated by the model comes at a cost: the
model fails to reproduce both the unemployment and vacant jobs volatilities. The matching
process acting like a labor adjustment cost, a large part of the adjustment is made through
prices (especially real wages) and therefore explains the low voatility of unemployment and
vancant jobs as well as the high volatility of labor productivity. The fact that the wage setting
rule, which implies a significant gap between wage and productivity, does not appear in the
moments can be explained by the low volatility of unemployment and of vacant jobs. By
inducing a low volatility of the labor market tightness, this lack of volatility implies that
productivity and wages are only slightly disconnected. The introduction of mismatch shocks
helps to improve the output of the model (see Langot 1994).
Part II: Problem (Exam 2003)
1/ The firm problem
We assume the existence of an effort supply function with the current wage wt and reference
wage w r as arguments : e  e~ ( w , w r ) . The reference wage is regarded as a function of the
t
t
t
t
wage paid by other firms wt , the income of self-employment b and the unemployment rate
u t : wtr  w r ( wt , b, ut ) .
a/ To what theory refers such an assumption? What drawback of the basic RBC framework do
you think such an assumption could remedy?
Answer:
Efficiency wage theory. It is supposed to improve the employment variability by explaining
variations in employment through a rationed labor supply rather than through workers'
preferences.
The representative firm maximises then the profit:  t  zt f (k t , nt e~ ( wt , wtr ))  rt k t  wt nt
b/ Assuming an interior solution, what are the first order conditions of the firm's problem ?
Answer:
zt (f / kt )  rt
z (f / n )e~ ( w , w r )  w
t
t
t
t
t
zt (f / nt )(e~ / wt )nt  nt
c/ Derive the Solow (1979) condition and explain what it means.
Answer:
e~( wt , wtr )
wt
1
~
wt
e ( wt , wtr )
At the optimum, the elasticity of effort with respect to wage has to be unitary

2/ The consumer problem
The representative consumer i program is :



 
max E   t V cit   G eit , wt , wtr X t
cit ,dkit ,eit
t 0
s.t.
cit  kit1  (r  1   )kit  wt  ut wt  b 
nit  n  1
where X t is a dummy variable which is equal to one if the particular individual under
consideration is employed, zero otherwise.
a/ Explain the utility function and the constraints
Answer:
The quantity of leisure or labor supplied does not enter as an argument of the representative
agent's utility function. Labor is inelastically supplied at a level normalized to unity. Effort is
an argument of the utility function. There is separability of the utility function in consumption
and effort. The dummy variable X implies that there is no utility or disutility from effort if the
agent is unemployed.
+ may be noted that the wage does not result from a Walrasian equilibriumbut is rather
determined by firms in the context of its impact on effort. This generates unemployment.
+ usual comments on the constraint
b/ What is the link between G eit , wt , wtr  and et  e~ ( wt , wtr ) ?
Answer:
et  e~ ( wt , wtr ) is a consequence of maximizing G eit , wt , wtr  with respect to eit .
3/ The equilibrium
a/ In which case is the economy in a Walrasian regime? What is then happening for the Solow
condition?
Answer:
In favorable states of nature, the demand for labor by firms at the optimally determined wage
may exceed the aggregate labor supply.We are then in a Walrasian regime with the wage set
at a level where labor demand and supply are balanced. This implies that the Solow condition
is not binding.
From now on, we assume that the economy lies entirely within the efficiency region.
We specify the following effort function and reservation wage:
e~(w / wr )  e n  M ln( w / wr )
w r  w1u b u
where e n is a constant giving the effort at the norm.
b/ Show that the firm optimally chooses the wage to stimulate a constant level of effort. What
is happening if the wage is such that the elasticity of effort with respect to wage is greater
than unity? By how much above the reference wage does the firm then sets wage ?
Answer:
The optimality condition   1 requires that the firm chooses the wage so that e  M .If
  1, the level of effort is then below M, and it pays to increase the wag rate until e  M .
We then have ln( w / wr )  (M  e n ) / M . Thus the firm sets its wage as a fixed percentage
above the reference wage where the percentage is equal to the percent excess effort above the
norm.
d/ Deduce a relationship between unemployment and the wage offered. What happens for the
wage offer as u approaches zero ? What does this imply for the labor demanded (look at the
first order conditions of the firm) and for the possibility to reach the walrasian regime under
the specification we have chosen?
Answer:
u(ln w  ln b)  (M  e n ) / M . As u approaches zero, the firm wants to increase its wage offer
without bound. The first order condition with respect to labor demanded implies that it leads
to a decrease in labor demanded. It follows that the Walrasian regime is not attainable under
this specification.
4/ Results
 x̂
 xˆ /  yˆ
 x̂
 xˆ /  yˆ
 x̂
 xˆ /  yˆ
ŷ
1.76
Table 1: Quarterly US time series
n̂
ĉ
iˆ
1.29
8.60
1.66
0.82
p̂
1.18
1
0.85
0.76
0.42
ŷ
2.15
Table 2: Equilibrium economy
n̂
ĉ
iˆ
1.24
13.89
0.35
p̂
1.79
1
0.804
1
1
ŷ
2.14
Table 3: Optimal economy
ĉ
iˆ
1.25
12.20
n̂
0.35
p̂
1.79
1
0.838
1
1
0.862
0.856
a/ Comment these results.
b/ How can you explain the result for labor volatility?
Answer:
Most of the adjustment is in terms of prices (wages) rather than of quantities (hours worked):
After a positive productivity shock, the labor marginal productivity increases. Thus, firms
increase their labor demand. The labor rate therefore rises which corresponds to a fall in
unemployment. Moreover, the Solow condition being satisfied, the real wage adjusts in order
to keep constant the effort supplied by agents. This implies that after a positive technological
shock, wages can rise to compensate the negative effect on effort of the potential fall in
unemployment
c/ What is your conclution as far as the contribution of this model to the RBC literature is
concerned? Do you have any suggestions to improve it?
Answer:
-not satisfactory to explain fluctuations on the labor market
-results largely depends on the form of the effort function that should be well justified
-it takes into account non walrasian behavior on the labor market
-it can be improved to take into account th temporal dimension in the effort function.